What is the term for a linear equation that does not lie in a single plane?

[RVM45]

...Can someone refresh my memory; what do you call a linear equation that does not lie in a single plane? Can you give me an example; or two? I come across the term in passing; in a book on Linear equations; that brought them up only to say that it wouldn't be covering them. I don't have the book anymore; and I've been trying to recall the term for some time.

...Wasn't sure whether to post this here; or in the Linear Equation sub-forum. Just gave it my best guess.

...RVM45

 

[HallsofIvy]

...Can someone refresh my memory; what do you call a linear equation that does not lie in a single plane? Can you give me an example; or two? I come across the term in passing; in a book on Linear equations; that brought them up only to say that it wouldn't be covering them. I don't have the book anymore; and I've been trying to recall the term for some time.

...Wasn't sure whether to post this here; or in the Linear Equation sub-forum. Just gave it my best guess.

...RVM45

Did you really come across "a linear equation that does not lie in a single plane" in a textbook? It makes no sense to talk about a "linear equation" lying or not lying in a plane, an equation is not a geometric object! I suspect that the reference was to a pair of lines that do not lie in a single plane. If two lines are parallel, then they line in one plane. If they cross, then they line if one plane. If two lines are skew, then they do not lie in a single plane. Is that the term you are looking for?

 

[coomast]

Can it be that you are referring to the equation of a straight line in space, therefore not lying in the classical 2D (XY) plane? This can be calculated using the intersection of two flat planes in space, which is indeed a straight line. The equation of a flat plane in space is:

$$A\cdot x+B\cdot y+C\cdot z+D=0$$

So, a straight line in space is therefore the solution to the system:

$$A_1\cdot x+B_1\cdot y+C_1\cdot z+D_1=0$$
$$A_2\cdot x+B_2\cdot y+C_2\cdot z+D_2=0$$

These are linear equations, therefore this idea. There are other ways of calculating this using vectors.

 

[RVM45]

...I mean like a hyperbola; parabola; bell curve; and a great many meandering curves all lie in one plane; and can be described with a 2-D; X-Y coordinate system. Now picture a parabola where at some point the line veers off rather abruptly into the Z axis- always a line- not a plane; plane segment; geometric solid; etc.; etc. But a curve that requires three variables to plot.

...At least that was how I understood it. The text was well over my head. I like mathematics texts; and when they're selling for a buck or so; at yard sales; or whatever- I'll buy them; even if I don't understand them.

...In the intro, the author brought up these curves; only to say that he wasn't going to get into them. He also mentioned a French word for such equations. Can't remember the first word; it looked vaguely like the English word "Curves" then if memory serves me correctly- De Gouche; or De Gauche.

...Maybe that will help clear it up. If not, perhaps I completely misunderstood.

...RVM45

 

[coomast]

Mmmm, not easy to find. I did some google work on "De Gauge" and "french mathematicians", and could only find one (possible) solution. The man called "Pierre Charles François Dupin" did work on curves in 3D.

The pictures I found on some sites seem to match the ones you are talking about. The site is:

http://www.lems.brown.edu/vision/people/leymarie/Notes/CurvSurf/Surfaces.html

I couldn't find anything on "De Gauge". Try google on "cyclides" as well. I also tried looking for Galois, he did work on higher order curves, but that didn't seem to get me anywhere.

Maybe this will bring you any closer.

 

[HallsofIvy]

Do you understand that what you have posted now has no relationship at all to your first post? You specifically asked about linear equations that do not lie in a single plane. The graph of a linear equation is a straight line, not a parabola or hyperbola or any such thing and must lie in a single plane. That was why I assumed you meant two lines. A curve that does not lie on a single plane must be given by a formula with non-zero torsion. That measures how the curve "twists" so that it cannot lie on a plane. A simple example is the "spiral" or helix:
x= cos(t), y= sin(t), z= t.

 

[RVM45]

...Sorry. I think that my terminology was once more precise; but it has been many years since I had a formal class; and I've gotten a bit- well more than a bit- careless with my terminology.

...I've always been fascinated by higher mth; despite the fact that I'm not very good at it. I have just a bit of Calculus; can handle some simple differentations. Never (yet) mastered integration.

...I am retired due to poor health; at the age of fifty. I've always wanted to learn more math. I have found forums to be good way to learn; and that's how I found this site. I mainly wanted to start with the simple stuff; and work my way up; but the 3-D curve question was something off-the-wall.

...My goal is to someday be able to fully understand String Theory; Rheiman's Conjector; and Quantum theory.

...At any rate...

...R

 

[RVM45]

...Sorry. I think that my terminology was once more precise; but it has been many years since I had a formal class; and I've gotten a bit- well more than a bit- careless with my terminology.

...I've always been fascinated by higher math; despite the fact that I'm not very good at it. I have just a bit of Calculus; can handle some simple differentations. Never (yet) mastered integration.

...I am retired due to poor health; at the age of fifty. I've always wanted to learn more math. I have found forums to be good way to learn; and that's how I found this site. I mainly wanted to start with the simple stuff; and work my way up; but the 3-D curve question was something off-the-wall.

...My goal is to someday be able to fully understand String Theory; Rheiman's Conjector; and Quantum theory.

...At any rate...

...RVM45

 

[coomast]

Don't mind the terminology RVM45, I often make mistakes with it. Anyway, good to hear that you have this profound interest in mathematics. We are here to help you for solving problems you might have.

It is not easy to look for a certain book if one does not know the title or author. Was it a French book? Do you have a picture of some kind showing a curve that you're referring to? It might help in obtaining a direction to look for. Dupin is not the one I suppose? Earlier today I stood in front of my bookshelf (approx. 350 books on physics, math, engineering) and was thinking which one could give me the answer. I don't have much French ones, most are Dutch and English, so it's not easy.

You have a long way to go if you want to understand String Theory, Riemann's conjecture and Quantum theory. Riemann is math, the other two are a part of physics but require some math to understand... Then again, you're in the right place to ask questions.